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Sage Quickstart for Abstract Algebra
====================================
This `Sage `_ quickstart tutorial was developed
for the MAA PREP Workshop "Sage: Using Open\-Source Mathematics Software
with Undergraduates" (funding provided by NSF DUE 0817071). It is
licensed under the Creative Commons Attribution\-ShareAlike 3.0 license
(`CC BY\-SA `_).
As computers are discrete and finite, anything with a discrete, finite
set of generators is natural to implement and explore.
Group Theory
-------------
Many common groups are pre-defined, usually as permutation groups:
that is, explicitly described as subgroups of symmetric groups.
- Every group of order 15 or less is available as a permutation group.
- Sometimes they are available under special names, though.
::
sage: G = QuaternionGroup()
sage: G
Quaternion group of order 8 as a permutation group
::
sage: H = AlternatingGroup(5)
sage: H
Alternating group of order 5!/2 as a permutation group
::
sage: H.is_simple()
True
::
sage: D = DihedralGroup(8)
sage: D
Dihedral group of order 16 as a permutation group
We can access a lot of information about groups, such as:
- A list of subgroups up to conjugacy,
- or a stabilizer,
- or other things demonstrated below.
::
sage: for K in D.conjugacy_classes_subgroups():
....: print K
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [()]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(1,5)(2,6)(3,7)(4,8)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(2,8)(3,7)(4,6)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(1,2)(3,8)(4,7)(5,6)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(2,8)(3,7)(4,6), (1,5)(2,6)(3,7)(4,8)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(1,2)(3,8)(4,7)(5,6), (1,5)(2,6)(3,7)(4,8)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(2,8)(3,7)(4,6), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(1,2,3,4,5,6,7,8), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(1,2)(3,8)(4,7)(5,6), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(2,8)(3,7)(4,6), (1,2,3,4,5,6,7,8), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)]
In the previous cell we once again did a for loop over a set of objects
rather than just a list of numbers. This can be very powerful.
::
sage: D.stabilizer(3)
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(1,5)(2,4)(6,8)]
::
sage: for K in D.normal_subgroups():
....: print K
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [()]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(1,5)(2,6)(3,7)(4,8)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(1,2)(3,8)(4,7)(5,6), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(2,8)(3,7)(4,6), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(1,2,3,4,5,6,7,8), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)]
Subgroup of (Dihedral group of order 16 as a permutation group) generated by [(1,2,3,4,5,6,7,8), (1,8)(2,7)(3,6)(4,5)]
We can access specific subgroups if we know the generators as a
permutation group.
::
sage: L = D.subgroup(["(1,3,5,7)(2,4,6,8)"])
::
sage: L.is_normal(D)
True
::
sage: Q=D.quotient(L)
sage: Q
Permutation Group with generators [(1,2)(3,4), (1,3)(2,4)]
::
sage: Q.is_isomorphic(KleinFourGroup())
True
There are some matrix groups as well, both finite and infinite.
::
sage: S = SL(2, GF(3))
sage: S
Special Linear Group of degree 2 over Finite Field of size 3
We can print out *all* of the elements of this group if we use a small Python trick we
haven't mentioned before. The string ``\n`` represents a new line.
::
sage: for a in S:
....: print a, "\n"
[1 0]
[0 1]
...
[2 2]
[2 1]
::
sage: SS = SL(2, ZZ)
Of course, you have to be careful what you try to do!
::
sage: SS.list()
Traceback (most recent call last):
...
NotImplementedError: group must be finite
::
sage: for a in SS.gens():
....: print a, "\n"
[ 0 1]
[-1 0]
...
Rings
------
Sage has many pre\-defined rings to experiment with. Here is how one
would access :math:`\ZZ/12\ZZ`, for instance.
::
sage: twelve = Integers(12)
sage: twelve
Ring of integers modulo 12
::
sage: twelve.is_field()
False
::
sage: twelve.is_integral_domain()
False
Quaternions, and generalizations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We can define generalized quaternion algebras, where :math:`i^2=a`,
:math:`j^2=b`, and :math:`k=i\cdot j`, all over :math:`\QQ`::
sage: quat = QuaternionAlgebra(-1, -1)
sage: quat
Quaternion Algebra (-1, -1) with base ring Rational Field
::
sage: quat.is_field()
False
::
sage: quat.is_commutative()
False
::
sage: quat.is_division_algebra()
True
::
sage: quat2 = QuaternionAlgebra(5, -7)
::
sage: quat2.is_division_algebra()
True
::
sage: quat2.is_field()
False
Polynomial Rings
~~~~~~~~~~~~~~~~
Polynomial arithmetic in Sage is a very important tool.
The first cell brings us back to the symbolic world. This is **not the
same thing** as polynomials!
::
sage: reset('x') # This returns x to being a variable
sage: (x^4 + 2*x).parent()
Symbolic Ring
Now we will turn :math:`x` into the generator of a polynomial ring. The
syntax is a little unusual, but you will see it often.
::
sage: R. = QQ[]
sage: R
Univariate Polynomial Ring in x over Rational Field
::
sage: R.random_element() # random
-5/2*x^2 - 1/4*x - 1
::
sage: R.is_integral_domain()
True
::
sage: (x^4 + 2*x).parent()
Univariate Polynomial Ring in x over Rational Field
::
sage: (x^2+x+1).is_irreducible()
True
::
sage: F = GF(5)
sage: P. = F[]
::
sage: P.random_element() # random
2*y
::
sage: I = P.ideal(y^3+2*y)
sage: I
Principal ideal (y^3 + 2*y) of Univariate Polynomial Ring in y over Finite Field of size 5
::
sage: Q = P.quotient(I)
::
sage: Q
Univariate Quotient Polynomial Ring in ybar over Finite Field of size 5 with modulus y^3 + 2*y
Fields
------
Sage has superb support for finite fields and extensions of the rationals.
Finite Fields
~~~~~~~~~~~~~
::
sage: F. = GF(3^4)
sage: F
Finite Field in a of size 3^4
The generator satisfies a Conway polynomial, by default, or the
polynomial can be specified.
::
sage: F.polynomial()
a^4 + 2*a^3 + 2
::
sage: F.list()
[0, a, a^2, a^3, a^3 + 1, a^3 + a + 1, a^3 + a^2 + a + 1, 2*a^3 + a^2 + a + 1, a^2 + a + 2, a^3 + a^2 + 2*a, 2*a^3 + 2*a^2 + 1, a^3 + a + 2, a^3 + a^2 + 2*a + 1, 2*a^3 + 2*a^2 + a + 1, a^3 + a^2 + a + 2, 2*a^3 + a^2 + 2*a + 1, 2*a^2 + a + 2, 2*a^3 + a^2 + 2*a, 2*a^2 + 2, 2*a^3 + 2*a, 2*a^3 + 2*a^2 + 2, a^3 + 2*a + 2, a^3 + 2*a^2 + 2*a + 1, 2*a^2 + a + 1, 2*a^3 + a^2 + a, a^2 + 2, a^3 + 2*a, a^3 + 2*a^2 + 1, a + 1, a^2 + a, a^3 + a^2, 2*a^3 + 1, 2*a^3 + a + 2, 2*a^3 + a^2 + 2*a + 2, 2*a^2 + 2*a + 2, 2*a^3 + 2*a^2 + 2*a, a^3 + 2*a^2 + 2, 2*a + 1, 2*a^2 + a, 2*a^3 + a^2, 2, 2*a, 2*a^2, 2*a^3, 2*a^3 + 2, 2*a^3 + 2*a + 2, 2*a^3 + 2*a^2 + 2*a + 2, a^3 + 2*a^2 + 2*a + 2, 2*a^2 + 2*a + 1, 2*a^3 + 2*a^2 + a, a^3 + a^2 + 2, 2*a^3 + 2*a + 1, 2*a^3 + 2*a^2 + a + 2, a^3 + a^2 + 2*a + 2, 2*a^3 + 2*a^2 + 2*a + 1, a^3 + 2*a^2 + a + 2, a^2 + 2*a + 1, a^3 + 2*a^2 + a, a^2 + 1, a^3 + a, a^3 + a^2 + 1, 2*a^3 + a + 1, 2*a^3 + a^2 + a + 2, a^2 + 2*a + 2, a^3 + 2*a^2 + 2*a, 2*a^2 + 1, 2*a^3 + a, 2*a^3 + a^2 + 2, 2*a + 2, 2*a^2 + 2*a, 2*a^3 + 2*a^2, a^3 + 2, a^3 + 2*a + 1, a^3 + 2*a^2 + a + 1, a^2 + a + 1, a^3 + a^2 + a, 2*a^3 + a^2 + 1, a + 2, a^2 + 2*a, a^3 + 2*a^2, 1]
::
sage: (a^3 + 2*a^2 + 2)*(2*a^3 + 2*a + 1)
2*a^3 + a^2 + a + 1
:math:`F` should be the splitting field of the polynomial
:math:`x^{81}-x`, so it is very good that we get no output from the
following cell, which combines a loop and a conditional statement.
::
sage: for a in F:
....: if not (a^81 - a == 0):
....: print "Oops!"
Field Extensions, Number Fields
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Most things you will need in an undergraduate algebra classroom are
already in Sage.
::
sage: N = QQ[sqrt(2)]
sage: N
Number Field in sqrt2 with defining polynomial x^2 - 2
::
sage: var('z')
z
sage: M.=NumberField(z^2-2)
sage: M
Number Field in a with defining polynomial z^2 - 2
::
sage: M.degree()
2
::
sage: M.is_galois()
True
::
sage: M.is_isomorphic(N)
True